Foundations — Parametric differentiation — dy - dx, d²y - dx²

This page is the ground floor. The parent topic freely uses symbols like $t$, $\dot{x}$, $\frac{dy}{dx}$, $\frac{d^{2}y}{d{x}^2}$ and words like slope, quotient rule, chain rule. Below, each one is built from nothing, in an order where every item leans only on the ones before it. If you can pass the checklist at the bottom, the parent note will read like a story you already know.

0. The point $(x,y)$ — where "a place on the plane" comes from

Look at the figure: the horizontal line is the $x$-axis (rightward = bigger $x$), the vertical line is the $y$-axis (upward = bigger $y$). Their crossing is the origin $(0,0)$. A point is found by walking $x$ across, then $y$ up.

Why the topic needs it: the whole game is describing a curve, and a curve is just a crowd of points $(x,y)$. Everything else is about how those points are chosen and how steep the trail between them is.

1. A function $x(t)$ — a machine that turns one number into another

Picture a vending machine: press button $t=2$, out drops a specific snack. Press $t=2$ again, you get the same snack — that reliability is what makes it a function.

Why the topic needs it: in parametric form we have two machines, $x(t)$ and $y(t)$, fed by the same input $t$. They together decide where the dot is.

2. The parameter $t$ — the hidden dial (think: time)

In the figure, as $t$ increases the orange dot marches around the circle. At $t=0$ it sits at the right; a quarter-turn later it's at the top. The same $x=0$ happens twice (top and bottom) — proof a plain $y=f(x)$ can't describe it, but $t$ can.

Why the topic needs it: without $t$ there is no "speed of $x$" or "speed of $y$" to compare — and that comparison is the slope.

3. Slope — "rise over run", the number that means steepness

The figure shows a right triangle riding on the line: the horizontal leg is the run, the vertical leg is the rise. Their ratio is the slope, and it stays the same no matter how big you draw the triangle — that constancy is what makes "slope" a single meaningful number.

Why the topic needs it: $\frac{dy}{dx}$ is slope. The entire first derivative is built to answer "how steep is the curve here?"

4. The symbol $\Delta$ and the shrink to $d$ — from a chunk to an instant

Picture zooming into the curve with a microscope. A finite $\Delta$-triangle straddles a bend and only approximates the steepness. Shrink it to a $d$-triangle and it hugs the curve so tightly it captures the steepness at a single point. This shrinking-to-a-point is the job of the limit — the reason this chapter is called Limits & Derivatives.

Why the topic needs it: $\frac{dy}{dx}$ is the slope of that infinitesimal triangle. The line "$dx=\dot{x}dt$" in the parent is just this differential idea applied through the dial $t$.

5. The derivative $\frac{dx}{dt}$ and the dot $\dot{x}$ — speed of a machine

Why the topic needs it: the first derivative is $\dot{y}/\dot{x}$ — the vertical speed divided by the horizontal speed. Everything hinges on these two dotted quantities.

6. The chain rule — how to differentiate through a middle-man

Why the topic needs it: the parent derives the slope formula by writing $\frac{dy}{dt}=\frac{dy}{dx}\cdot \frac{dx}{dt}$ and solving for $\frac{dy}{dx}$. No chain rule, no formula. Full details live in Chain rule.

7. The quotient rule — differentiating a fraction

Why the topic needs it: the second derivative differentiates the fraction $\frac{\dot{y}}{\dot{x}}$. The parent applies exactly this rule with $u=\dot{y}$, $v=\dot{x}$, producing the $\ddot{y}\dot{x}-\dot{y}\ddot{x}$ on top and ${\dot{x}}^2$ underneath — before one final divide by $\dot{x}$ gives the famous ${\dot{x}}^3$. More at Quotient rule.

8. The second derivative $\frac{d^{2}y}{d{x}^2}$ — how the slope itself bends

In the figure, a curve bending like a smile (holding water) has increasing slope — that's positive $\frac{d^{2}y}{d{x}^2}$, called concave up. A frown shape has decreasing slope — negative, concave down. The sign is the whole point, explored in Concavity and second derivative.

Why the topic needs it: it's the second headline formula. Its correct derivation demands the chain rule and the quotient rule and the differential idea — every prerequisite above, stacked.

How the foundations feed the topic

Equipment checklist

Cover the right side; say the answer aloud before revealing.

Connections

• Chain rule — the multiply-the-rates engine behind both formulas.
• Quotient rule — differentiates the fraction $\dot{y}/\dot{x}$ for the second derivative.
• Implicit differentiation — a cousin technique when there is no parameter.
• Tangents and normals — uses the slope you build here.
• Concavity and second derivative — reads the sign of $\frac{d^{2}y}{d{x}^2}$.
• Parametric curves, Cycloid — where these symbols come to life.